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Pseudo-Differential Operators

Pseudodifferential Methods in Number Theory

Authors: Unterberger, Andre

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  • Explores a new approach to the Riemann hypothesis
  • Explains the link between the theory of modular distributions and the classical one of modular forms
  • Includes previously unpublished material 
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About this book

Classically developed as a tool for partial differential equations, the analysis of operators known as pseudodifferential analysis is here regarded as a possible help in questions of arithmetic. The operators which make up the main subject of the book can be characterized in terms of congruence arithmetic. They enjoy a Eulerian structure, and are applied to the search for new conditions equivalent to the Riemann hypothesis. These consist in the validity of certain parameter-dependent estimates for a class of Hermitian forms of finite rank. The Littlewood criterion, involving sums of Möbius coefficients, and the Weil so-called explicit formula, which leads to his positivity criterion, fit within this scheme, using in the first case Weyl's pseudodifferential calculus, in the second case Fuchs'. 

The book should be of interest to people looking for new possible approaches to the Riemann hypothesis, also to new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.

Reviews

“The book is devoted to applications of pseudodifferential calculus to analytic number theory, aimed to new approaches to the Riemann hypothesis (RH). ... The book by A. Unterberger will be interesting and useful both for number theorists looking for new techniques, and for specialists in pseudodifferential operators interested in new application areas.” (Anatoly N. Kochubei, zbMath 1411.11004, 2019)


Table of contents (8 chapters)

Table of contents (8 chapters)

Buy this book

eBook $59.99
price for USA in USD
  • ISBN 978-3-319-92707-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $79.99
price for USA in USD
  • ISBN 978-3-319-92706-0
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Pseudodifferential Methods in Number Theory
Authors
Series Title
Pseudo-Differential Operators
Series Volume
13
Copyright
2018
Publisher
Birkhäuser Basel
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-92707-7
DOI
10.1007/978-3-319-92707-7
Softcover ISBN
978-3-319-92706-0
Series ISSN
2297-0355
Edition Number
1
Number of Pages
VI, 173
Topics

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